Finite Automata and Randomness
Conference paper
First Online:
Abstract
The lecture surveys approaches using finite automata to define several notions of (automata-theoretic) randomness.
It focuses on the one hand on automata-theoretic randomness of infinite sequences in connection with automata-independent notions like disjunctivity and Borel normality.
On the other hand it considers the scale of relaxations of randomness (Borel normality and disjunctivity), that is, finite-state dimension and subword complexity and their interrelations.
Keywords
Finite automata Infinite words Betting automata Finite-state dimension Subword complexityReferences
- 1.Ambos-Spies, K., Busse, E.: Automatic forcing and genericity: on the diagonalization strength of finite automata. In: Calude, C.S., Dinneen, M.J., Vajnovszki, V. (eds.) DMTCS 2003. LNCS, vol. 2731, pp. 97–108. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-45066-1_7CrossRefzbMATHGoogle Scholar
- 2.Becher, V., Carton, O., Heiber, P.A.: Normality and automata. J. Comput. Syst. Sci. 81(8), 1592–1613 (2015)MathSciNetCrossRefGoogle Scholar
- 3.Becher, V., Heiber, P.A.: Normal numbers and finite automata. Theor. Comput. Sci. 477, 109–116 (2013)MathSciNetCrossRefGoogle Scholar
- 4.Bourke, C., Hitchcock, J.M., Vinodchandran, N.: Entropy rates and finite-state dimension. Theor. Comput. Sci. 349(3), 392–406 (2005)MathSciNetCrossRefGoogle Scholar
- 5.Calude, C.S.: Information and Randomness. An Algorithmic Perspective. Texts in Theoretical Computer Science. An EATCS Series, 2nd edn. Springer, Berlin (2002). https://doi.org/10.1007/978-3-662-04978-5. With forewords by Gregory J. Chaitin and Arto SalomaaCrossRefGoogle Scholar
- 6.Dai, J.J., Lathrop, J.I., Lutz, J.H., Mayordomo, E.: Finite-state dimension. Theor. Comput. Sci. 310(1–3), 1–33 (2004)MathSciNetCrossRefGoogle Scholar
- 7.Doty, D., Lutz, J.H., Nandakumar, S.: Finite-state dimension and real arithmetic. Inf. Comput. 205(11), 1640–1651 (2007)MathSciNetCrossRefGoogle Scholar
- 8.Doty, D., Moser, P.: Finite-state dimension and lossy decompressors. CoRR abs/cs/0609096 (2006). http://arxiv.org/abs/cs/0609096
- 9.Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer, New York (2010)CrossRefGoogle Scholar
- 10.Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and Its Applications. Texts and Monographs in Computer Science. Springer, New York (1993). https://doi.org/10.1007/978-1-4757-3860-5
- 11.Martin-Löf, P.: The definition of random sequences. Inf. Control 9, 602–619 (1966)MathSciNetCrossRefGoogle Scholar
- 12.Moldagaliyev, B., Staiger, L., Stephan, F.: On the values for factor complexity (2018, to appear)CrossRefGoogle Scholar
- 13.Nies, A.: Computability and Randomness, Oxford Logic Guides, vol. 51. Oxford University Press, Oxford (2009). https://doi.org/10.1093/acprof:oso/9780199230761.001.0001
- 14.Oxtoby, J.C.: Measure and Category, Graduate Texts in Mathematics, vol. 2, 2nd edn. Springer, New York (1980). A survey of the analogies between topological and measure spacesGoogle Scholar
- 15.Perrin, D., Pin, J.E.: Infinite Words. Automata, Semigroups, Logic and Games. Elsevier/Academic Press, Amsterdam (2004)Google Scholar
- 16.Schnorr, C.P.: Zufälligkeit und Wahrscheinlichkeit. LNM, vol. 218. Springer, Heidelberg (1971). https://doi.org/10.1007/BFb0112458CrossRefzbMATHGoogle Scholar
- 17.Schnorr, C.P., Stimm, H.: Endliche Automaten und Zufallsfolgen. Acta Inf. 1, 345–359 (1972)MathSciNetCrossRefGoogle Scholar
- 18.Sheinwald, D., Lempel, A., Ziv, J.: On compression with two-way head machines. In: Storer, J.A., Reif, J.H. (eds.) Proceedings of the IEEE Data Compression Conference, DCC 1991, Snowbird, Utah, 8–11 April 1991, pp. 218–227. IEEE Computer Society (1991). https://doi.org/10.1109/DCC.1991.213359
- 19.Sheinwald, D., Lempel, A., Ziv, J.: On encoding and decoding with two-way head machines. Inf. Comput. 116(1), 128–133 (1995)MathSciNetCrossRefGoogle Scholar
- 20.Shen, A.: Automatic Kolmogorov complexity and normality revisited. In: Klasing, R., Zeitoun, M. (eds.) FCT 2017. LNCS, vol. 10472, pp. 418–430. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-55751-8_33CrossRefGoogle Scholar
- 21.Staiger, L.: Reguläre Nullmengen. Elektron. Informationsverarbeit. Kybernetik 12(6), 307–311 (1976)MathSciNetzbMATHGoogle Scholar
- 22.Staiger, L.: Kolmogorov complexity and Hausdorff dimension. Inform. Comput. 103(2), 159–194 (1993). https://doi.org/10.1006/inco.1993.1017MathSciNetCrossRefGoogle Scholar
- 23.Staiger, L.: \(\omega \)-languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 339–387. Springer, Berlin (1997). Beyond WordsCrossRefGoogle Scholar
- 24.Staiger, L.: Rich \(\omega \)-words and monadic second-order arithmetic. In: Nielsen, M., Thomas, W. (eds.) CSL 1997. LNCS, vol. 1414, pp. 478–490. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0028032CrossRefGoogle Scholar
- 25.Tadaki, K.: Phase transition and strong predictability. In: Ibarra, O.H., Kari, L., Kopecki, S. (eds.) UCNC 2014. LNCS, vol. 8553, pp. 340–352. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08123-6_28CrossRefzbMATHGoogle Scholar
- 26.Terwijn, S.A.: Complexity and randomness. Rend. Semin. Mat. 62(1), 1–37 (2004). TorinoMathSciNetzbMATHGoogle Scholar
- 27.Thomas, W.: Automata on infinite objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 133–191. Elsevier, Amsterdam (1990). Formal Models and SemanticsGoogle Scholar
- 28.Trakhtenbrot, B.A., Barzdiń, Y.M.: Finite Automata. North-Holland Publishing Co., Amsterdam (1973). Behavior and Synthesis, Translated from the Russian by D. Louvish, English translation edited by E. Shamir and L. H. Landweber, Fundamental Studies in Computer Science, vol. 1Google Scholar
- 29.Varacca, D., Völzer, H.: Temporal logics and model checking for fairly correct systems. In: 21th IEEE Symposium on Logic in Computer Science (LICS 2006), 12–15 August 2006, Seattle, WA, USA, Proceedings, pp. 389–398. IEEE Computer Society (2006). https://doi.org/10.1109/LICS.2006.49
- 30.Völzer, H., Varacca, D.: Defining fairness in reactive and concurrent systems. J. ACM 59(3), Article no. 13, 37 (2012). https://doi.org/10.1145/2220357.2220360
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